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Online Least-Squares Density Difference


The online Least Squares Density Difference detector is a non-parametric method for online drift detection. The LSDD between two distributions \(p\) and \(q\) on \(\mathcal{X}\) is defined as

\[LSDD(p,q) = \int_{\mathcal{X}} (p(x)-q(x))^2 \,dx\]

and also has an empirical estimate \(\widehat{LSDD}(\{X_i\}_{i=1}^N, \{Y_i\}_{i=t}^{t+W})\) that can be updated at low cost as the test window is updated to \(\{Y_i\}_{i=t+1}^{t+1+W}\). The detector is motivated by, but is a modified version of, Bu et al. (2017).

Online detectors assume the reference data is large and fixed and operate on single data points at a time (rather than batches). These data points are passed into the test-window and a two-sample test-statistic (in this case an estimate of LSDD) between the reference data and test-window is computed at each time-step. When the test-statistic exceeds a preconfigured threshold, drift is detected. Configuration of the thresholds requires specification of the expected run-time (ERT) which specifies how many time-steps that the detector, on average, should run for in the absence of drift before making a false detection. It also requires specification of a test-window size, with smaller windows allowing faster response to severe drift and larger windows allowing more power to detect slight drift.

For high-dimensional data, we typically want to reduce the dimensionality before passing it to the detector. Following suggestions in Failing Loudly: An Empirical Study of Methods for Detecting Dataset Shift, we incorporate Untrained AutoEncoders (UAE), black-box shift detection using the classifier’s softmax outputs (BBSDs) and PCA as out-of-the box preprocessing methods. Preprocessing methods which do not rely on the classifier will usually pick up drift in the input data, while BBSDs focuses on label shift.

Detecting input data drift (covariate shift) \(\Delta p(x)\) for text data requires a custom preprocessing step. We can pick up changes in the semantics of the input by extracting (contextual) embeddings and detect drift on those. Strictly speaking we are not detecting \(\Delta p(x)\) anymore since the whole training procedure (objective function, training data etc) for the (pre)trained embeddings has an impact on the embeddings we extract. The library contains functionality to leverage pre-trained embeddings from HuggingFace’s transformer package but also allows you to easily use your own embeddings of choice. Both options are illustrated with examples in the Text drift detection on IMDB movie reviews notebook.




  • x_ref: Data used as reference distribution.

  • ert: The expected run-time (ERT) in the absence of drift.

  • window_size: The size of the sliding test-window used to compute the test-statistic. Smaller windows focus on responding quickly to severe drift, larger windows focus on ability to detect slight drift.

Keyword arguments:

  • backend: Backend used for the LSDD implementation and configuration.

  • preprocess_x_ref: Whether to already preprocess and store the reference data.

  • preprocess_fn: Function to preprocess the data before computing the data drift metrics.

  • sigma: Optionally set the bandwidth of the Gaussian kernel used in estimating the LSDD. Can also pass multiple bandwidth values as an array. The kernel evaluation is then averaged over those bandwidths. If sigma is not specified, the ‘median heuristic’ is adopted whereby sigma is set as the median pairwise distance between reference samples.

  • n_bootstraps: The number of bootstrap simulations used to configure the thresholds. The larger this is the more accurately the desired ERT will be targeted. Should ideally be at least an order of magnitude larger than the ERT.

  • n_kernel_centers: The number of reference samples to use as centers in the Gaussian kernel model used to estimate LSDD. Defaults to 2*window_size.

  • lambda_rd_max: The maximum relative difference between two estimates of LSDD that the regularization parameter lambda is allowed to cause. Defaults to 0.2 as in the paper.

  • input_shape: Shape of input data.

  • data_type: Optionally specify the data type (tabular, image or time-series). Added to metadata.

Additional PyTorch keyword arguments:

  • device: Device type used. The default None tries to use the GPU and falls back on CPU if needed. Can be specified by passing either ‘cuda’, ‘gpu’ or ‘cpu’. Only relevant for ‘pytorch’ backend.

Initialized drift detector example:

from import LSDDDriftOnline

cd = LSDDDriftOnline(x_ref, ert, window_size, backend='tensorflow')

The same detector in PyTorch:

cd = LSDDDriftOnline(x_ref, ert, window_size, backend='pytorch')

We can also easily add preprocessing functions for both frameworks. The following example uses a randomly initialized image encoder in PyTorch:

from functools import partial
import torch
import torch.nn as nn
from import preprocess_drift

device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')

# define encoder
encoder_net = nn.Sequential(
    nn.Conv2d(3, 64, 4, stride=2, padding=0),
    nn.Conv2d(64, 128, 4, stride=2, padding=0),
    nn.Conv2d(128, 512, 4, stride=2, padding=0),
    nn.Linear(2048, 32)

# define preprocessing function
preprocess_fn = partial(preprocess_drift, model=encoder_net, device=device, batch_size=512)

cd = LSDDDriftOnline(x_ref, ert, window_size, backend='pytorch', preprocess_fn=preprocess_fn)

The same functionality is supported in TensorFlow and the main difference is that you would import from import preprocess_drift. Other preprocessing steps such as the output of hidden layers of a model or extracted text embeddings using transformer models can be used in a similar way in both frameworks. TensorFlow example for the hidden layer output:

from import HiddenOutput, preprocess_drift

model = # TensorFlow model; tf.keras.Model or tf.keras.Sequential
preprocess_fn = partial(preprocess_drift, model=HiddenOutput(model, layer=-1), batch_size=128)

cd = LSDDDriftOnline(x_ref, ert, window_size, backend='tensorflow', preprocess_fn=preprocess_fn)

Check out the Online Drift Detection on the Wine Quality Dataset example for more details.

Alibi Detect also includes custom text preprocessing steps in both TensorFlow and PyTorch based on Huggingface’s transformers package:

import torch
import torch.nn as nn
from transformers import AutoTokenizer
from import preprocess_drift
from alibi_detect.models.pytorch import TransformerEmbedding

device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')

model_name = 'bert-base-cased'
tokenizer = AutoTokenizer.from_pretrained(model_name)

embedding_type = 'hidden_state'
layers = [5, 6, 7]
embed = TransformerEmbedding(model_name, embedding_type, layers)
model = nn.Sequential(embed, nn.Linear(768, 256), nn.ReLU(), nn.Linear(256, enc_dim)).to(device).eval()
preprocess_fn = partial(preprocess_drift, model=model, tokenizer=tokenizer, max_len=512, batch_size=32)

# initialise drift detector
cd = LSDDDriftOnline(x_ref, ert, window_size, backend='pytorch', preprocess_fn=preprocess_fn)

Again the same functionality is supported in TensorFlow but with from import preprocess_drift and from alibi_detect.models.tensorflow import TransformerEmbedding imports.

Detect Drift

We detect data drift by sequentially calling predict on single instances x_t (no batch dimension) as they each arrive. We can return the test-statistic and the threshold by setting return_test_stat to True.

The prediction takes the form of a dictionary with meta and data keys. meta contains the detector’s metadata while data is also a dictionary which contains the actual predictions stored in the following keys:

  • is_drift: 1 if the test-window (of the most recent window_size observations) has drifted from the reference data and 0 otherwise.

  • time: The number of observations that have been so far passed to the detector as test instances.

  • ert: The expected run-time the detector was configured to run at in the absence of drift.

  • test_stat: LSDD metric between the reference data and the test_window if return_test_stat equals True.

  • threshold: The value the test-statsitic is required to exceed for drift to be detected if return_test_stat equals True.

preds = cd.predict(x_t, return_test_stat=True)

Resetting the detector with the same reference data and thresholds but with a new and empty test-window is straight-forward: